Mallory:
. . . would the Stable Master, Mallory, or Rhydd (since he's made the highest Animal Handling check thus far) reasonably know of any way to manage the egg in a way that an Axe Beak would find acceptable?
The answer is probably as straight forward as it gets; you'll have to bring the egg with you, trust Frida understands the egg is being moved, and then juggle it so Frida gets time to incubate. It would probably render Frida unavailable for long distance travel (more than 4-5 hours), unless you take breaks in-between. To lift the veil even further, it is stated plainly in the module that sled dogs require breaks every single hour or they gain a level of exhaustion. Which is . . . a strange way to handle it? Dogs can definitely run for a few hours at a time, but they'll require a proportionately long rest.
As to any specific nesting information on Axe Beaks, the wiki is scarce, so we'll have to infer from other big birds in real life, such as the ostrich or emu. Dodos were also somewhat large, speaking of flightless birds. The incubation period is about 42 days for ostriches, 56 for emus, and it is said to be 49 for dodos. Let's put it around 50 days.
From some quick google searching, it seems like one of two things are going to happen if you move the egg from its current nest; Frida is going to abandon it, or she is going to keep incubating it. Either way you should be good, right? Whether or not Frida will become pissy is another matter, but if anything has been established at this point it is that she is mostly cooperative. That is, as long as she realizes the egg has been moved, which could be found out by some quick trial and error. Since we need to expedite things we can handle this OOC, and since it is fully within my jurisdiction, I'm just going to go ahead and say you're good.
Dannika would be able to help you bring the egg to Bryn Shander since she has a bag of holding (you wrap it in some cloth first to pad it), but if you wish to use Frida for other distances you should be prepared to carry it on your own. If all of this is too much I suggest you just go with Wifni. That option is definitely on the table, don't let me try to talk you into anything!